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Second-order sufficient conditions for control problems with mixed control-state constraints

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Abstract

References 1–4 develop second-order sufficient conditions for local minima of optimal control problems with state and control constraints. These second-order conditions tighten the gap between necessary and sufficient conditions by evaluating a positive-definiteness criterion on the tangent space of the active constraints. The purpose of this paper is twofold. First, we extend the methods in Refs. 3, 4 and include general boundary conditions. Then, we relate the approach to the two-norm approach developed in Ref. 5. A direct sufficiency criterion is based on a quadratic function that satisfies a Hamilton-Jacobi inequality. A specific form of such a function is obtained by applying the second-order sufficient conditions to a parametric optimization problem. The resulting second-order positive-definiteness conditions can be verified by solving Riccati equations.

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Authors and Affiliations

  1. Institut für Numerische und Instrumentelle Mathematik, Westfälische Wilhelms Universität Münster, Münster, Germany

    H. Maurer (Professor)

  2. Institut für Mathematik, Technische Universität Cottbus, Cottbus, Germany

    S. Pickenhain (Professor)

Authors
  1. H. Maurer
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  2. S. Pickenhain
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Communicated by R. Bulirsch

The authors wish to thank K. Malanowski for helpful discussions.

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Maurer, H., Pickenhain, S. Second-order sufficient conditions for control problems with mixed control-state constraints. J Optim Theory Appl 86, 649–667 (1995). https://doi.org/10.1007/BF02192163

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